Mathematics and Computation in Imaging Science and Information Processing July-December, 2003 Organized by Institute of Mathematical Sciences and Center for Wavelet. Approximation, and Information Processing, National University of Singapore. Collaboration with the Wavelet Center for Ideal Data Representation. Co-chairmen of the organizing committee: Amos Ron (UW-Madison), Zuowei Shen (NUS), Chi-Wang Shu (Brown University)

Conferences Wavelet Theory and Applications: New Directions and Challenges, July 2003 Numerical Methods in Imaging Science and Information Processing, December 2003

Confirmed Plenary Speakers for Wavelet Conference Albert Cohen Wolfgang Dahmen Ingrid Daubechies Ronald DeVore David Donoho Rong-Qing Jia Yannis Kevrekidis Amos Ron Peter Schröder Gilbert Strang Martin Vetterli

Workshops IMS-IDR-CWAIP Joint Workshop on Data Representation, Part I on 9 – 11, II on July 2003 Functional and harmonic analyses of wavelets and frames, 28 July - 1 Aug 2003 Information processing for medical images, September 2003 Time-frequency analysis and applications, September 2003 Mathematics in image processing, December 2003 Industrial signal processing (TBA) Digital watermarking (TBA)

Tutorials A series of tutorial sessions covering various topics in approximation and wavelet theory, computational mathematics, and their applications in image, signal and information processing. Each tutorial session consists of four one-hour talks designed to suit a wide range of audience of different interests. The tutorial sessions are part of the activities of the conference or workshop associated with.

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Wavelet Algorithms for High-Resolution Image Reconstruction Zuowei Shen Department of Mathematics National University of Singapore Joint work with (accepted by SISC) T. Chan (UCLA), R.Chan (CUHK) and L.X. Shen (WVU)

Part I: Problem Setting Part II: Wavelet Algorithms Outline of the talk

What is an image? image = matrix pixel intensity = matrix entry Resolution = size of the matrix

I. High-Resolution Image Reconstruction: Resolution = 64  64Resolution = 256  256

Four low resolution images (64  64) of the same scene. Each shifted by sub-pixel length. Construct a high-resolution image (256  256) from them.

#2 #4 Boo and Bose (IJIST, 97): #1 taking lens CCD sensor array relay lenses partially silvered mirrors

Four 2  images merged into one 4   image: a1a1 a2a2 a3a3 a4a4 b1b1 b2b2 b3b3 b4b4 c1c1 c2c2 c3c3 c4c4 d1d1 d2d2 d3d3 d4d4 Four low resolution images Observed high- resolution image a1a1 b1b1 a2a2 b2b2 c1c1 d1d1 c2c2 d2d2 a3a3 b3b3 a4a4 b4b4 c3c3 d3d3 c4c4 d4d4 By permutation

Four 64  64 images merged into one by permutation: Observed high- resolution image by permutation

Modeling Consider: Low-resolution pixel High-resolution pixels Observed image: HR image passing through a low-pass filter a. LR image: the down samples of observed image at different sub-pixel position.

L f = g, After modeling and adding boundary condition, it can be reduced to : Where L is blurring matrix, g is the observed image and f is the original image.

The problem L f = g is ill-conditioned. g Here R can be I, . It is called Tikhonov method ( or the least square ) Regularization is required:

Wavelet Method Let â be the symbol of the low-pass filter. Assume: can be found such that One can use unitary extension principle to obtain a set of tight frame systems.

Let  be the refinable function with refinement mask a, i.e. Let  d be the dual function of  : We can express the true image as where v(  ) are the pixel values of the high-resolution picture.

The pixel values of the observed image are given by The observed function is The problem is to find v(  ) from (a * v)(  ). From 4 sets low resolution pixel values reconstruct f, lift 1 level up. Similarly, one can have 2 level up from 16 set...

Do it in the Fourier domain. Note that We have or

Generic Wavelet Algorithm: (i) Choose (ii) Iterate until convergence: Proposition Suppose that and nonzero almost everywhere. Then for arbitrary.

Regularization: Damp the high-frequency components in the current iterant. Wavelet Algorithm I: (i) Choose (ii) Iterate until convergence:

Matrix Formulation: The Wavelet Algorithm I is the stationary iteration for Different between Tikhonov and Wavelet Models: L d instead of L *. Wavelet regularization operator. Both penalize high-frequency components uniformly by .

Wavelet Thresholding Denoising Method: Decompose the n-th iterate, i.e., into different scales: ( This gives a wavelet packet decomposition of n-th iterate.) Denoise these coefficients of the wavelet packet by thresholding method. Before reconstruction,

Wavelet Algorithm II: (i) Choose (ii) Iterate until convergence: Where T is a wavelet thresholding processing.

4  4 sensor array: Original LR FrameObserved HR Tikhonov Algorithm I Algorithm II

4  4 sensor array: Tikhonov Algorithm II

2  2 sensor array: 1 level up 4  4 sensor array: 2 level up Numerical Examples:

1-D Example: Signal from Donoho’s Wavelet Toolbox. Blurred by 1-D filter. Original Signal Observed HR Signal Tikhonov Algorithm II

Ideal low-resolution pixel position High-resolution pixels Calibration Error: Problem no longer spatially invariant. Displaced low- resolution pixel Displacement error   x

The lower pass filter is perturbed The wavelet algorithms can be modified

Reconstruction for 4  4 Sensors: (2 level up) Original LR FrameObserved HR Tikhonov Wavelets

Reconstruction for 4  4 Sensors: (2 level up) Tikhonov Wavelets

Numerical Results: 2  2 sensor array (1 level up) with calibration errors: 4  4 sensor array (2 level) with calibration errors:

(0,0) (1,1) (0,2) (1,3) (2,0) (3,1) (2,2) (3,3) (0,1)(0,3) (1,0) (2,1) (1,2) (2,3) (3,0)(3,2) Example: 4  4 sensor with missing frames: Super-resolution: not enough frames

(0,1)(0,3) (1,0) (2,1) (1,2) (2,3) (3,0)(3,2) Example: 4  4 sensor with missing frames: Super-resolution: not enough frames

i.Apply an interpolatory subdivision scheme to obtain the missing frames. ii.Generate the observed high-resolution image w. iii.Solve for the high-resolution image u. iv.From u, generate the missing low-resolution frames. v.Then generate a new observed high-resolution image g. vi.Solve for the final high-resolution image f. Super-Resolution: Not enough low-resolution frames.

Reconstructed Image: Observed LR Final Solution